Let $r$, $s$, and $t$ be solutions of the equation $x^3-5x^2+6x=9$.

Compute $\frac{rs}t + \frac{st}r + \frac{tr}s$.
Solution: The cubic equation can be rewritten as $x^3-5x^2+6x-9=0$. First, we list the relations that Vieta's formulas give us:
\begin{align*}
-(r+s+t) &= -5,\quad(\clubsuit) \\
rs+rt+st &= 6,\phantom{-}\quad(\textcolor{red}{\diamondsuit}) \\
-rst &= -9.\,\quad(\textcolor{red}{\heartsuit})
\end{align*}We wish to compute
$$\frac{rs}t + \frac{rt}s + \frac{st}r = \frac{r^2s^2+r^2t^2+s^2t^2}{rst}.$$The denominator is $rst=9$. To obtain the numerator, we square equation $(\textcolor{red}{\diamondsuit})$ to get
$$r^2s^2 + r^2t^2 + s^2t^2 + 2r^2st + 2rs^2t + 2rst^2 = 36.$$We can rewrite this as
$$r^2s^2 + r^2t^2 + s^2t^2 + 2rst(r+s+t) = 36.$$From equations $(\clubsuit)$ and $(\textcolor{red}{\heartsuit})$, we have
$$2rst(r+s+t) = 2\cdot 9\cdot 5 = 90,$$so
$$r^2s^2 + r^2t^2 + s^2t^2 = 36 - 90 = -54.$$Finally, we have
$$\frac{rs}t + \frac{rt}s + \frac{st}r = \frac{r^2s^2 + r^2t^2 + s^2t^2}{rst} = \frac{-54}{9} = \boxed{-6}.$$